Document Stieltjes constants

[cheuberg]

According to https://en.wikipedia.org/wiki/Stieltjes_constants as well as
to http://dlmf.nist.gov/25.2.E4, the Laurent series expansion of the Riemann
zeta function around s=1 is

zeta(s) = 1/(s-1) + gamma_0 - gamma_1 (s-1) + O((s-1)^2).

However, sage says

sage: zeta(s).series(s==1, 2)
1*(s - 1)^(-1) + (euler_gamma) + (-1/2*stieltjes(1))*(s - 1) + Order((s - 1)^2)

(note the denominator 2 in the coefficient of (s-1)).

Numerically,

sage: stieltjes(1.)
-0.0728158454836767

and Wikipedia states that

gamma_1 = −0.0728158454836767248605863758749013191377363383 	 ...

It seems that there is an incorrect factor in sage (this is now fixed and doctested).

When fixing this, the documentation of stieltjes should be improved to actually contain a definition of the stieltjes constants instead of simply stating that "The Stieltjes constants are used in the series expansions of zeta(s)."

Depends on #21855

CC: @behackl

Component: symbolics

Issue created by migration from https://trac.sagemath.org/ticket/21899

[cheuberg]

Description changed:

--- 
+++ 
@@ -2,23 +2,29 @@
 to http://dlmf.nist.gov/25.2.E4, the Laurent series expansion of the Riemann
 zeta function around s=1 is
 
+```
 zeta(s) = 1/(s-1) + gamma_0 - gamma_1 (s-1) + O((s-1)^2).
+```
 
 However, sage says
 
+```
 sage: zeta(s).series(s==1, 2)
 1*(s - 1)^(-1) + (euler_gamma) + (-1/2*stieltjes(1))*(s - 1) + Order((s - 1)^2)
-
+```
 (note the denominator 2 in the coefficient of (s-1)).
 
 Numerically,
 
+```
 sage: stieltjes(1.)
 -0.0728158454836767
-
+```
 and Wikipedia states that
 
+```
 gamma_1 = −0.0728158454836767248605863758749013191377363383 	 ...
+```
 
 It seems that there is an incorrect factor in sage.
 

[rwst]

comment:2

See #21963 where I fixed the doctests. I thought it better to use this ticket for the documentation fixes.

[behackl]

comment:3

Replying to @rwst:

See #21963 where I fixed the doctests. I thought it better to use this ticket for the documentation fixes.

Thank you Ralf! We'll polish the documentation of Stieltjes-constants on this ticket.

[cheuberg]

comment:4

I suggest to have one doctest with slightly increased precision for zeta around 1, just to see more than one stieltjes constant.

[cheuberg]

Dependencies: #21855

[cheuberg]

Upstream: Fixed upstream, in a later stable release.

[rwst]

Description changed:

--- 
+++ 
@@ -26,7 +26,7 @@
 gamma_1 = −0.0728158454836767248605863758749013191377363383 	 ...
 ```
 
-It seems that there is an incorrect factor in sage.
+It seems that there is an incorrect factor in sage (this is now fixed and doctested).
 
 When fixing this, the documentation of stieltjes should be improved to actually contain a definition of the stieltjes constants instead of simply stating that "The Stieltjes constants are used in the series expansions of zeta(s)."
 

[rwst]

Changed upstream from Fixed upstream, in a later stable release. to none